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arXiv:1404.2442 · Physical Review A

Gapless topological Fulde-Ferrell superfluidity induced by an in-plane Zeeman field

Hui Hu, Lin Dong, Ye Cao, Han Pu, Xia-Ji Liu

Published Sep 24, 2014• 23 citations
arXivHTMLPDF

Abstract

Topological superfluids are recently discovered quantum matter that hosts topologically protected gapless edge states known as Majorana fermions---exotic quantum particles that act as their own antiparticles and obey non-Abelian statistics. Their realizations are believed to lie at the heart of future technologies such as fault-tolerant quantum computation. To date, the most efficient scheme to create topological superfluids and Majorana fermions is based on the Sau-Lutchyn-Tewari-Das Sarma model with a Rashba-type spin-orbit coupling on the x-y plane and a large out-of-plane (perpendicular) Zeeman field along the z direction. Here we propose an alternative setup, where the topological superfluid phase is driven by applying an in-plane Zeeman field. This scheme offers a number of different features, notably Cooper pairings at finite center-of-mass momentum (i.e., Fulde-Ferrell pairing) and gapless excitations in the bulk. As a result, gapless topological quantum matter with an inhomogeneous pairing order parameter appears. It features unidirectional Majorana surface states at boundaries, which propagate in the same direction and connect two Weyl nodes in the bulk. We demonstrate the emergence of such exotic topological matter and the associated Majorana fermions in spin-orbit coupled atomic Fermi gases, and we determine its parameter space. The implementation of our scheme in semiconductor/superconductor heterostructures is briefly discussed.

DOI
10.1103/physreva.90.033624

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The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. The very recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with proper spin-orbit coupling and Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram and the edge modes in this new topological matter are presented. The momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman field and chemical potential. The in-plane Zeeman field and finite momentum pairing induce asymmetry to the effective $p$-wave pairing, apart from a small fraction of higher orbital components. The phase diagram is composed by different phases, which are determined by the topology and band gap nature of the superfluids. Especially, the gapped and gapless topological FFLO phase have totally different finite size effect. These novel features show that the spin-orbit coupled cold atoms provides an important platform in realizing topological matters which may not be materialized with solids.

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Topological superfluids are recently discovered quantum matter that hosts topologically protected gapless edge states known as Majorana fermions---exotic quantum particles that act as their own antiparticles and obey non-Abelian statistics. Their realizations are believed to lie at the heart of future technologies such as fault-tolerant quantum computation. To date, the most efficient scheme to create topological superfluids and Majorana fermions is based on the Sau-Lutchyn-Tewari-Das Sarma model with a Rashba-type spin-orbit coupling on the x-y plane and a large out-of-plane (perpendicular) Zeeman field along the z direction. Here we propose an alternative setup, where the topological superfluid phase is driven by applying an in-plane Zeeman field. This scheme offers a number of different features, notably Cooper pairings at finite center-of-mass momentum (i.e., Fulde-Ferrell pairing) and gapless excitations in the bulk. As a result, gapless topological quantum matter with an inhomogeneous pairing order parameter appears. It features unidirectional Majorana surface states at boundaries, which propagate in the same direction and connect two Weyl nodes in the bulk. We demonstrate the emergence of such exotic topological matter and the associated Majorana fermions in spin-orbit coupled atomic Fermi gases, and we determine its parameter space. The implementation of our scheme in semiconductor/superconductor heterostructures is briefly discussed.
We theoretically investigate a three-dimensional Fermi gas with Rashba spin–orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde–Ferrell pairing, gapped or gapless topological order, and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space (i.e., massless Dirac fermions). The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin–orbit coupling. In this context, topological superfluid phase with an imperfect Rashba spin–orbit coupling is also studied.
Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metallic at the boundary. Here we report that a gapless, topologically nontrivial superfluid with an inhomogeneous Fulde-Ferrell pairing order parameter can emerge in a two-dimensional spin-orbit coupled Fermi gas, in the presence of both in-plane and out-of-plane Zeeman fields. The Fulde-Ferrell pairing-induced by the spin-orbit coupling and in-plane Zeeman field-is responsible for this gapless feature. This exotic superfluid has a significant Berezinskii-Kosterlitz-Thouless transition temperature and has robust Majorana edge modes against disorder owing to its topological nature.
Topological Fermi superfluids have played the central role in various fields of physics. However, all previous studies focus on the cases where Cooper pairs have zero center-of-mass momenta (i.e., normal superfluids). The topology of Fulde–Ferrell (FF) superfluids with nonzero momentum pairings have never been explored until recent findings that FF superfluids in a spin–orbit (SO) coupled Fermi gas can accommodate Majorana fermions in real space in low dimensions and Weyl fermions in momentum space in three dimension. In this review, we first discuss the mechanism of pairings in SO coupled Fermi gases in optical lattices subject to Zeeman fields, showing that SO coupling as well as Zeeman fields enhance FF states while suppress Larkin–Ovchinnikov states. We then present the low temperature phase diagram including both FF superfluids and topological FF superfluids phases in both two dimension and three dimension. In one dimension, Majorana fermions as well as phase dependent order parameter are visualized. In three dimension, we show the properties of Weyl fermions in momentum space such as anisotropic linear dispersion, Fermi arch, and gaplessness away from k⊥ = 0. Finally, we discuss some possible methods to probe FF superfluids and topological FF superfluids in cold atom systems.
The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. Recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with appropriate spin-orbit coupling and in-plane Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram, and the edge modes in this topological matter are presented. The center-of-mass momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman fields and chemical potential. The in-plane Zeeman fields also induce asymmetry to the effective ppp-wave pairing, apart from a small fraction of higher orbital components. As a result, the phase diagram is composed by different phases, which are determined by the Fermi surface topology and band gap width of the superfluids. Especially, the gapped and gapless topological FFLO phases have totally different finite size effects. These features show that the spin-orbit coupled cold atoms provide an important platform in realizing topological matters which may not be materialized with solids.
We theoretically investigate the superfluid density and Berezinskii-Kosterlitz-Thouless (BKT) transition of a two-dimensional Rashba spin-orbit-coupled atomic Fermi gas with both in-plane and out-of-plane Zeeman fields. It was recently predicted that, by tuning the two Zeeman fields, the system may exhibit different exotic Fulde-Ferrell (FF) superfluid phases, including the gapped FF, gapless FF, gapless topological FF, and gapped topological FF states. Due to the FF paring, we show that the superfluid density (tensor) of the system becomes anisotropic. When an in-plane Zeeman field is applied along the xxx direction, the tensor component along the yyy direction ns,yy{n}_{s,yy}ns,yy​ is generally larger than ns,xx{n}_{s,xx}ns,xx​ in most parameter space. At zero temperature, there is always a discontinuity jump in ns,xx{n}_{s,xx}ns,xx​ as the system evolves from a gapped FF into a gapless FF state. With increasing temperature, such a jump is gradually washed out. The critical BKT temperature has been calculated as functions of the spin-orbit-coupling strength, interatomic interaction strength, and in-plane and out-of-plane Zeeman fields. We predict that the novel FF superfluid phases have a significant critical BKT temperature, typically at the order of 0.1TF0.1{T}_{F}0.1TF​, where TF{T}_{F}TF​ is the Fermi degenerate temperature. Therefore, their observation is within the reach of current experimental techniques in cold-atom laboratories.
The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. The very recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with proper spin-orbit coupling and Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram and the edge modes in this new topological matter are presented. The momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman field and chemical potential. The in-plane Zeeman field and finite momentum pairing induce asymmetry to the effective ppp-wave pairing, apart from a small fraction of higher orbital components. The phase diagram is composed by different phases, which are determined by the topology and band gap nature of the superfluids. Especially, the gapped and gapless topological FFLO phase have totally different finite size effect. These novel features show that the spin-orbit coupled cold atoms provides an important platform in realizing topological matters which may not be materialized with solids.